Homological algebra for the representation Green functor for abelian groups
HTML articles powered by AMS MathViewer
- by Joana Ventura PDF
- Trans. Amer. Math. Soc. 357 (2005), 2253-2289 Request permission
Abstract:
In this paper we compute some derived functors $\operatorname {Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product. When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor $\operatorname {Ext}$. When the group is $G=\mathbb {Z}/2\times \mathbb {Z}/2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired $\operatorname {Ext}$ functors.References
- Serge Bouc, Green functors and $G$-sets, Lecture Notes in Mathematics, vol. 1671, Springer-Verlag, Berlin, 1997. MR 1483069, DOI 10.1007/BFb0095821
- Andreas W. M. Dress, Contributions to the theory of induced representations, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 183–240. MR 0384917
- L. Gaunce Lewis, Jr. The box product of Mackey functors. Unpublished.
- L. Gaunce Lewis, Jr. The theory of Green functors. Unpublished notes, 1981.
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
- Jacques Thévenaz and Peter Webb, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865–1961. MR 1261590, DOI 10.1090/S0002-9947-1995-1261590-5
- Jacques Thévenaz and Peter J. Webb, Simple Mackey functors, Proceedings of the Second International Group Theory Conference (Bressanone, 1989), 1990, pp. 299–319. MR 1068370
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Additional Information
- Joana Ventura
- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: jventura@math.ist.utl.pt
- Received by editor(s): August 22, 2003
- Published electronically: May 10, 2004
- Additional Notes: The author was partially supported by FCT grant Praxis XXI/BD/11357/97 and a one year research grant from Calouste Gulbenkian Foundation
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2253-2289
- MSC (2000): Primary 55P91, 18G10
- DOI: https://doi.org/10.1090/S0002-9947-04-03566-4
- MathSciNet review: 2140440